Cooperative Diversity in Interference Limited Wireless Networks

Transcription

1 Cooperative Diverity in Interference Limited Wirele Network Sam Vakil, Student Member, IEEE, and Ben Liang, Senior Member, IEEE Abtract Uing relay in wirele network can potentially lead to ignificant capacity increae. However, within an aynchronou multi-uer communication etting, relaying might caue more interference in the network, and ignificant um-rate deterioration may be oberved. In thi work the effect of cooperation in an interference limited, narrow-band wirele network i invetigated. It i crucial to determine the optimal trade-off between the amount of throughput gain obtained via cooperation and the amount of interference introduced to the network. We quantify the amount of cooperation uing the notion of a cooperative region for each active node. The node which lie in uch a region are allowed to cooperate with the ource. We adopt the decode-and-forward cheme at the relay and ue the phyical interference model to determine the probability that a relay node correctly decode it correponding ource. Through numerical analyi and imulation, we tudy the optimal cooperative region ize to maximize the network um-rate and energy efficiency, baed on network ize, relay availability, node decoding threhold, and detinatioeception capability. It i hown that optimized ytem performance in term of the network um-rate and the power efficiency i ignificantly improved compared with cae where relay node are not exploited or where the cooperative region ize i uboptimal. Index Term Cooperative diverity, um-rate, interference, cooperation gain, cooperative region. I. INTRODUCTION Spatial multiplexing gain a a mean to increae the capacity ha been extenively tudied in the context of multiple-inputmultiple-output (MIMO) ytem []. Although the ue of multiple-antenna i appealing in theory, in ome application uch a wirele enor network, it might not be feaible to benefit from thi degree of freedom in ytem deign due to the limited ize and computing capability of an individual node. However, in uch dene environment, uing the reource of other peer node can help improve the network performance. In thi context, relay node can be exploited a a mean to increae the capacity in a wirele network. The relay channel firt introduced by van der Meulen lead to a communication cheme where intead of point-to-point communication between the ource and detination, relay are exploited in a two-hop communication. The key capacity reult for the cae of a ingle relay were introduced by Cover and El Gamal in [2]. The capacity region for the relay channel with M relay i not known to date. However, Gatpar and Vetterli [3] have obtained upper and lower bound on the capacity caling under Gauian noie and how that thee bound meet in the limit of large number of node. The author are with the Department of Electrical and Computer Engineering, Univerity of Toronto. {vakil, liang}@comm.utoronto.ca A counterpart of MIMO ytem ha beeecently propoed in which the capacity and diverity gain of MIMO ytem are obtained via ditributed antenna. Thi cheme, called cooperative diverity, i propoed a a mean to combat fading. It i an intereting concept in multi-uer communication introduced mainly by Laneman et al [4][5] and Sendonari et al [6]. Thi problem ha it root in the two-hop relay problem which ha been an open problem in information theory. Under thi etting, the relay channel i ued to forward the data cauing an increae in the capacity pecifically for the cae where the ource-detination channel experience deep fade. Nabar et al [7] further evaluate the performance of cooperative cheme in the cae of ingle ource, ingle relay, and ingle detination and prove that full diverity can be obtained. In [8] Sankaranarayanan, Kramer and Mandayam further conider the cae where multiple ource end their meage to a relay, and the relay either imply forward the data or firt decode and then forward. Although relay problem remain unolved in term of the optimal communication cheme, the tudie above have taken ignificant tep in quantifying the performance gain obtained from cooperation. However, reearch in thi ubject ha been mainly concentrated on information theoretic conideration, and only a few tudie have focued on the ytem level performance of cooperation uch a [9], in which Mergen et al tudy the problem of cooperative broadcat under a ingle-ource ingle-detination etup with multiple level of cooperation, where the broadcat performance i quantified by finding a ignal decoding threhold above which a meage i propagated in the network. The few prior work which conider multiple ource do not quantify the interference within the network and uually aume there are a et of rule (uch a TDM or FDM cheduling), which lead to interference removal. However, for the general non-orthogonal cae where multiple ource and relay node ue the ame channel, their tranmiion will interfere with each other. In a narrow-band wirele network with multiple ource, interference greatly affect the network capacity and ha to be conidered. In thi work, we tudy the effect of relaying trategie and cooperation in uch a multipleource network, where the node employ un-ynchronized tranmiion. We ak the baic quetion How hould the node optimally balance cooperation and interference to maximize the network capacity? Extending our preliminary work in [], we addre thi quetion more thoroughly and conider a wider range of important metric for node communication performance including um-rate and power aving. To the bet of our knowledge, there i no other work

2 2 quantifying the trade-off between cooperation and interference in the literature. The cloet work to our i perhap [], where Quek et al tudy a dene wirele enor network, in which the node end one information bit about the exitence of a Phenomenon of Interet (PoI) to a fuion center. The author compare the performance metric uch a the network energy conumption and correct decoding probability at the fuion center of a parallel fuion architecture with the cooperative fuion architecture (CFA). The author how trade-off exit between patial diverity gain, average energy conumption, delivery ratio of the conenu flooding protocol, network connectivity, node denity, and PoI intenity in CFA. Specifically, CFA i advantageou in cae with weak PoI intenity. In our work, although we tudy the potential energy gaieulting from cooperation, our focu i on a network with multiple ource and relay. By quantifying the added interference in the network reulting from cooperation, we addre the quetion of Whether or not to cooperate? from a different view point. Dardari et al conider a wirele enor network in [2] and tudy the interdependent apect of WSN communication protocol and ignal proceing. They tudy the trade-off between energy conervation and the amount of error in the etimation of a calar field by uing appropriate ditributed ignal proceing method, which i different from our focu on the added interference a a reult of cooperation and it effect on the network energy gain. The main contribution of thi work are three fold. Firt, we preent a general network architecture for localized regionbaed cooperation in a large wirele network with multiple ource. Second, we propoe an analytical framework to invetigate the relation between the cooperative regioadiu, the interference level, and the relay decoding probability, and we derive the network um-rate given multiple antenna at the detination a a main metric for cooperative region optimization, in a MIMO multiple acce etting. Third, we demontrate the power aving obtained via cooperation under a wide range of activity level at the node, and we evaluate the effect of different decoding threhold on the network performance. Our numerical and imulation tudie provide general deign guideline for optimal relaying in an interference limited network. The ret of thi paper i organized a follow. Section II explain the network model and preent a practical relaying algorithm. Section III explain the detail of our analytical framework to tudy the interaction betweeelaying and interference, which i reflected in the derivation of the decoding probability for each node. In Section IV we compute the network um-rate with relaying, baed on the reult of Section III, on deriving an etimate for the number of ucceful relay, which cooperate with each ource node. Section V preent the numerical and experimental reult. Finally, concluding remark are given in Section VI. II. WIRELESS NETWORK MODEL AND RELAYING SCHEME In thi ection, we explain the network model under conideration and preent a generic relaying architecture. A. Network Model and Data Diemination We conider a collection of N node placed randomly, uniformly and independently in a dik A. For illutration, we aume a communication etting imilar to SENMA [3], in which the node on the dik need to communication with a mobile acce point, which may be the data ink if the network model repreent a enor network. We call it the Acce Point (AP) throughout thi paper. The AP i a powerful unit both in it proceing capability and it ability to travere the network and i located at a variable height h, < h < above the center of the dik. Time i aumed to be lotted to interval of length L equal to the length of a data packet. Each node in the dik incur an activity event at each lot. In a enor network, thi could mean that the node meaure a phyical phenomenon in cae it ene activity. The activity event of a node at any lot i modeled by an independent Bernoulli random variable u i for node i, { p, if u p(u i ) = i =, if u i =. () We denote by X i [k] the data packet ent by node i in cae of activity at time k. The random proce X i i aumed to be i.i.d among the ource node. Thi aumption i motivated by the ue of cooperative region to be explained later, which allow eparation between the ource node o that their activitie may be independent. The AP i aumed to have receive antenna, while the regular node each ha only one antenna. Since the AP ha the capability of interference mitigation uing multiple antenna, our focu of interference analyi i on node in the dik. Figure give a imple illutration for a poible direct uplink communication cheme and our propoed cooperative counterpart, which will be dicued in the next ection. Pathlo and channel variation are both conidered in the channel model. When node i tranmit with power P i [n], node j receive the tranmiion with power P i [n]γ ij [n]. The channel gain can be repreented a γ ij = h ij 2, (2) where r ij i the ditance between node i and j and h ij model the fading channel from node i to j and α i the path lo rolling factor. Throughout the paper a block Rayleigh fading channel i conidered for which the channel gain i contant over a block of length L. In our imulatioeult, we alo conider the effect of increaing the line of ight component by uing a Rician fading model between the node and the AP. B. Cooperative Scheme Under the propoed cheme, the node are divided to three group in term of their operation. The firt group are the active ource node, which are the node that have a packet to end at the beginning of a lot. The actual et of permiible ource will be choen among thee node by the cheduling algorithm, which i explained in detail in the next ection. The econd group, choen among the remaining node, comprie r α ij

3 3 the et of potential relay, which try to cooperate with the ource. Thee potential relay try to implement a decode and forward cheme. The main goal of our work i to find the optimal et of the potential relay and permiible ource, which reult in the maximum network um-rate from the ource node toward the AP. We further quantify the relay node, a ub-et of the potential relay, which are indeed ucceful in the decoding of their intended meage. The lat et of the node comprie the one not choen among the active ource and potential relay. Thee node hould not attempt to relay and mut remain ilent during the communication. We define the ith cooperative region, C i, a the area in which node i act a a ource and it potential relay are contained. We denote it radiu by r C (Fig. ). The optimal value for thi radiu will be determined by the value which maximize the network um-rate. Relay node in each uch region uually do not have the capability of imultaneou tranmiion and reception. Therefore, their tranmiion are aumed to be half-duplex. The communication of a meage from the ource node i divided into two tep. The ource i, after ening the environment, firt broadcat the meage X i [k] to the AP and the potential relay in it cooperative region. A potential relay node can etimate it memberhip in the cooperative region of active ource node in it vicinity by etimating it relative ditance to the ource, poibly by uing the received ignal power. In the econd phae, the potential relay which have uccefully decoded the meage will forward it to the detination. A we will how later, to maximize the umrate, the optimal cooperative region will not cover the whole network but i limited to a mall area around each permitted ource. Since the ucceful decoding relay deliver the ame meage X i [k], they act a a multi-antenna ytem ending a common meage toward the AP. The two-phae communication i depicted in Fig.. During each lot a different et of ource are activated. Therefore, their correponding relay are different. For intance, a hown in Fig., uppoe the relay from node j cooperative region were in their reception phae during the lat time lot and are now in the cooperative phae. Then they are the firt caue of interference to the potential relay in node i cooperative region. The ource node which have tarted their tranmiion ynchronouly with ource i are the econd caue of interference. Source l in Fig. repreent uch an example. Uing the ame phyical model a the one introduced in [4], a relay m i aumed to uccefully decode the meage ent from the ource i if the SINR at m i above a required threhold. The phyical model requirement for the decoding at relay m can be written a One of the key challenge to implementing relay-baed cooperation protocol i block and ymbol ynchronization of the cooperating terminal. Such ynchronization might be obtained through periodic tranmiion of known ynchronization prefixe [4]. A detailed tudy involving the ynchronization iue i beyond the cope of thi work. SINR m = P i [k]γ im [k] N + l S[k],l i P l[k]γ lm [k] + j C[k ] P j[k]γ jm [k] > β, where the interference at the potential relay node during the relay reception phae i either due to other permitted ource node compriing the et of the active ource, S[k], at time k, or the relay which have completed their reception in lot k and are forwarding their correponding meage to AP in lot k compriing the et C[k ] = N p[k ] i= C i [k ], where N p [k] repreent the number of permitted ource (cooperative region) at time k. The noie i aumed to be a complex Gauiaandom variable with variance N. Due to operation in the high interference regime reulting from multiple imultaneou communication, we neglect the noie effect and bae our analyi on SIR at the relay. The parameter β i a deign parameter and depend on the level of tolerated interference by the node. Since we conider fading in our model, the relay within each region are till probable to go under deep fade. Therefore, ome of the relay may not be capable of decoding uccefully. However, the cloer a relay node i to it correponding ource the higher the decoding probability i. We define two tate for each potential relay, receive and tranmit. During the receive tate the ignal received at a potential relay i decoded correctly if (3) hold. If a potential relay i in the cooperative region of a ource, and it ha uccefully decoded the meage from the ource, it tranmit the meage to the AP in the tranmit tate. During the relay tranmiion tate we have the following expreion Y m [k] =, Y d [k] = N p[k] i= (3) H i X i [k] + Z, (4) where Y m [k] i the received meage at a relay when it i in the tranmit tate, Y d [k] i the received vector of dimenion at the detination, which i a uperpoition of the meage ent by all zone, H i i the channel vector from the et of relay iegion i which have uccefully decoded the meage to the AP in addition to ource i itelf, X i [k] repreent the meage vector ent out ynchronouly by the relay of region i and ource i, and Z i zero-mean complex Gauian noie with independent, equal-variance real and imaginary part. C. Source Scheduling In the propoed cheme, within each cooperative region only one ource i allowed to tranmit during each communication cycle. We focu on a naphot of the network at time k. At time k, the potential relay have tried to decode their intended meage, and among them, the et of the ucceful decoding relay, D[k ], forward their meage toward the AP during the lot k. The timing of thee two lot ha been depicted in Fig. 2. Clearly, in lot k, the ource allowed to tranmit hould be choen among the node which were not

4 4 AP AP Coop. region l at time n Cooperative Region i at time n Tranition to Cooperative Phae Coop. Region j at time n- Fig.. Network layout. Tranition from direct tranmiion phae to cooperative communication: the relay that have decoded their meage in time lot n are interfering with the relay which are in their receive tate in time n. i { Fig. 2. Timing for a ource and it correponding relay in C i a well a poible interferer ource and relay. choen in the prior lot among the ource, S[k ], or relay, D[k ]. Auming that we know the optimal cooperative region area a = πrc, 2 to be determined a a reult of our optimization, we can characterize the ultimate number of ource that are permitted to imultaneouly tranmit a A a, where A i the coverage area. In practice thi number trongly depend on the network topology. Two ource i and j can imultaneouly end their meage if they do not lie in the ame cooperative region. The problem of finding the et of imultaneou ource can therefore be tranlated into the maximal independent et (MIS) problem by conidering a graph where the vertex et repreent the active ource node and there i an edge between two vertice i and j if and only if their ditance d i,j < 2r C. In our experiment, we have implemented the parallel algorithm preented in [5] to olve the MIS problem and found the maximal packing number. During each iteration of thi algorithm, the active ource node can communicate locally to determine if their ditance from the ource node already choen in the previou iteration i maller than 2r C. Under thi etting the communication cheme within each cooperative region can be modeled a a ingle ource and multiple relay. III. CHARACTERIZATION OF THE INTERFERENCE AND DECODING RELAYS IN COOPERATIVE REGIONS In thi ection we quantify the interference and the correct decoding probability at a relay a a function of the ignal t t t t t to interference ratio at the relay, for a given ize of the cooperative region. Clearly, the amount of interference at a relay i a function of the number of interferer and their relative location to the relay. Thee interferer either belong to the et C[k ] or the et S[k] a we explained in Section II- C. During the ource tranmiion phae, a relay m C i [k] decode the correponding ource if and only if (3) hold. The overall interference at m can then be expreed a I m [k] = Pγ lm [k] + Pγ jm [k] l S[k],l i l= j C[k ] N p D = Pγ lm [k] + Pγ jm [k], j= where D i i the number of node which have been ucceful in decoding within C i [k] and D = Np i= D i repreent the total number of interfering relay, and we have implified the problem of relay election by auming that the node tranmit with the ame power P k = P. We next formulate the cooperative region maximization problem and analytically find the expected number of ucceful decoding relay within each region. A. Expected Number of Succeful Decoding Relay We denote the number of relay iegion i by the random variable NC i. Node are uniformly ditributed over the dik area. Therefore, the event m C i ha a Bernoulli ditribution with Pr[m C i ] = ai A, where a i = πr 2 C. NC i follow a binomial ditribution a Pr[NC i = l] = ( ) N l ( a i A )l ( ai A )N l with mean E[NC i ] = N ai A = Nπ r2 C A. In the following we quantify the expected number of ucceful decoding relay. Propoition : In the given wirele network withiegion i, E[D i ] = r C N A Pr[SIR(r) > β]2πrdr, where Pr[SIR(r) > β] repreent the ucceful decoding probability for a relay located at ditance r relative to it ource. Proof: The proof i imilar to the proof of Theorem I in [] (with the light difference that we have to replace E[Nr] i by E[N i C ] = Nπ r2 C A in (9) of []). We can further compute the expected number of total ucceful decoding relay within the network. Propoition 2: In the given wirele network the expected number of total ucceful decoding node during each time lot atifie E[D] = E[N p ]E[D i ]. (5)

5 5 Proof: Refer to Appendix I for the proof. B. Expected Number of Interfering Node Since the node are randomly ditributed on the dik, the number of interferer node i a random variable. In time lot k the relay which have been ucceful in their reception phae in lot k interfere with the relay receiving in the current lot. The total number of interfering relay can therefore be formulated a N Irelay [k] = N p[k] j= N j I relay [k], (6) where N j I relay [k] repreent the number of ucceful decoding relay within cooperative region j (which have been ucceful ieceive mode during lot k ). Due to the ymmetry of the communication tructure in different lot, the number of interferer i tationary and we remove the time dependence in the expectation. An upper bound for the number of circular cooperative region that can be packed in the dik area equal A πr 2 C a explained in Section II-C. Furthermore, the event of being an active ource i Bernoulli with probability and the expected number of active ource equal N. Since thi number can not exceed the above limit, we conclude E[N p ] = min( N, A πr 2 C ). Uing the conditional expected value law, we can write [ E[N Irelay ] = E Np E[NIrelay N p = n p ] ] [ Np = E Np j= E[N j I relay ] ]. Baed on our interference analyi in Section III-C it will be hown that the geographic location of the cooperative region only lightly affect the amount of interference at each relay. However, the relative location of the relay to it correponding ource i the main factor which affect the amount of interference. Uing thi ymmetry and the fact that the interferer relay are the relay that are tranmitting a meage which ha been uccefully decoded during the previou time lot, we can conclude that the expected number of interferer relay in any region i equal to that of another region j. We therefore can write E[N Irelay ] = E Np [ (Np )E[N j I relay ] ] = (E[N p ] )E[N j I relay ]. (7) Finally, the total number N I [k] of interfering node i the um of N Irelay and the number of ource in the current lot which are interfering with relay m reception. Since the ource correponding to the relay ha to be removed from the et of interfering ource, we have E[N I ] = E[N Irelay ] + E[N p ]. C. Interference Analyi: A Continuum Approach In thi ection we explain the detail for deriving an approximation to the amount of interference at each relay. The channel from each interfering node l to the relay m i aumed to undergo Rayleigh fading. We further aume that the magnitude of fading i contant for each packet (quaitatic fading). To find a cloed from expreion for the amount y" x". h d j c.. Fig. 3. Snaphot of a dik and a cooperative region. A ource and it relay m i hown. Differential element of the dik area have been conidered to find the overall interference at m by integrating over thi area. of interference, we ue a continuum type approach imilar to the one ued in [6] in crux. Figure 3 demontrate a poible cooperative region (the mall circle). The area outide thi circle i the potential interference region. For each interfering node l, the amount of interference to node m equal I lm = P r h lm α lm 2. We conider Rayleigh fading. Therefore h lm 2 ha exponential ditribution with parameter, and hence E[I lm ] = µ l = P r. lm α We conider a wirele network with high node denity ρ. In uch a continuum model we can ue a differential approach to evaluate the expected value of interference at m. Since there are a total of N I [k] interfering node, in the limit of a large number of node, in each differential element we have N I A rdrdθ node. Therefore, an element dθ a depicted in Fig. 3 on average caue the following amount of interference di = P r α N I rdrdθ = A m y ' ' x P N I r α A drdθ = ρ P drdθ. (8) rα The overall expected interference at node m i, therefore, E[I m ] = di, where the integration i performed over S, S the potential interferer region. For a differential element located at angle θ with repect to x in Fig. 3, the egment mh repreent the ditance of the maximum interferer, d max (θ). Here, x i the axi in the direction cm. The egment mj i the ditance of m from the minimum poible interferer, and it i repreented a d min (ω), where ω i the angle of mj with x defined in the direction of m (thee coordinate are ued to find the equation for the two circle). The derivation of d min (ω) and d max (θ) i explained in Section III-C [] for the intereted reader. The overall expected interference at m can be formulated a E[I m ] = = ρp α 2 2π dmax(θ) d min( θ+ x mx ) 2π ρ P drdθ rα [ d min ( θ + x mx ) α 2 ]dθ. d max (θ) α 2 (9) In general, numerical integration i needed to olve the above integral. However, in our analyi we have conidered

6 6 Interference power magnitude at the relay Relay dik center ditance ource relay ditance Fig. 4. The amount of interference (normalized by the number of interferer) at a relay veru relay location in dik and it locatioelative to it ource. r c =.5 and A =. α = 4. For thi cae it can be hown that (more detailed explanation i given in [], p.g. 5) E[I m ] = 2 ρp ( A π 2 α 2 π 2 d 4 (c,m) 2π A d 2 (c,m) + A 2 πrc 2 ). d 4 (,m) 2d 2 (,m)rc 2 + r4 C () E[I m] Figure 4 repreent the normalized value, N I, of the expected interference within the dik of unit area, a a function of the ditance between the dik center c and the relay m. Alo, the effect of the relay locatioelative to it correponding ource, within each cooperative region, ha been conidered. Note that interetingly the change of the ditance relative to the dik center doe not caue ubtantial change in the expected interference value. However, within each region, a poible relay that i cloer to the region boundarie undergoe a higher amount of interference a expected. D. Succeful Decoding Probability In thi ection we will give an approximation for Pr[SIR m > β]. We conider equal power P = for all node. The interference at relay m can be formulated a I m = NI j= Ij m, where each interference element ha an exponential ditribution with mean µ l a we explained in Section III-C. We replace N I by it expected value a an approximation, and aume that individual interference element have equal mean µ = E[Im] E[N. Then, we have the um of E[N I] I] i.i.d exponential random variable with mean µ, which ha Erlang ditribution with parameter E[N I ] and mean µe[n I ] = E[I m ]. Hence, f Im (x) = x E[NI] (E[N I ] )!µ E[NI] e x µ, for x. () The ditribution of the ignal power received at node m located at ditance d from the correponding ource can.5 then be computed and the detail are given in Appendix II. The maieult of thi ection therefore can be tated a Corollary : We can formulate the outage probability for each relay a Pr[SIR m β] = F Y (β) = ( ). + µβ E[NI] (2) µ Here µ = P d i a function of the node relative ditance α d with it correponding ource. The expected number of decoding node within each cooperative region can therefore be formulated by the reult of Propoition and i a function it radiu. In the next ection we ue thi reult to find the optimal area for the cooperative region, in which the relay are allowed to decode and forward. To implify the analyi we aume that thi area i a circle and the radiu of thi circle i the ame for all active node. The latter i jutified by the fact that the average amount of interference at each relay i not enitive to the relative location of the cooperative region within the planar dik, while the ditance of the relay from it correponding ource i the determining factor. The metric of interet in thi cae i the network um-rate. IV. NETWORK SUM-RATE OPTIMIZATION In Section III we introduced an analytical framework to quantify the number of ucceful decoding relay. To derive the overall network um-rate, we conider the following twopart contribution of data flow toward the AP. In the firt part, the et of active ource cheduled to tranmit end their meage toward the detination. Thi phae can be modeled a multiple-acce communication. In the econd part, the et of ucceful decoding relay forward the decoded meage to the AP a depicted in Fig.. In thi phae, the ucceful decoding relay contitute a cooperative MIMO ytem. The relay node within each cooperative region C i erve a the multiple antenna ending a common meage ynchronouly. The network um-rate during thee two phae can be written a R Total = 2 (R Ph + R Ph2 ), (3) where R Ph i the um rate during the firt non-cooperative phae and R Ph2 i the um-rate during the econd phae when cooperation i in effect. The cooperative regioadiu optimization can be written a r opt = arg max r C E[R Ph2 ] = arg max r C N p E[R i (D i )] i= ubject to m D i SIR m > β and i,j d(i,j) > 2r opt, (4) where we ue the notation R Ph2 = N p i= R i(d i ) to clarify that R i (D i ), the data rate correponding to region i, i baed on having D i node in thi region. Thi optimization i contrained by the fact that any relay m within a region ha to atify the SIR requirement to correctly decode. Alo, the ditance requirement impoed by cheduling ha to be atified.

7 7 The above optimization problem i non-convex i C, o we ue the following approximation r opt arg max r C N p R i (E[D i ]). (5) i= Thi approximation arie ince the expected value of a concave function f(x) obey E[f(x)] f[e[x]] baed on Jenen inequality. The inequality become tight a the concavity decreae and the comparion between our reult from analyi and imulation ugget that the bound i indeed tight. Therefore, the choice of the cooperative region which reult in the maximum expected number of decoding relay, E[D i ], will maximize the network um-rate. The problem in thi cae i eaier to olve ince E[D i ] can be computed uing the reult of Propoition and (2) in term of r C (noting that N I i a function of r C ). We olve thi optimization problem numerically. The capacity of a MIMO channel ha been derived in the landmark work of Telatar []. Next, we further clarify the multiuer MIMO model and how that our etting follow the ame cheme, auming that the AP ha acce to the channel tate information. The number of tranmit antenna in each region i equal to the number of ucceful decoding node, approximated by E[D i ]. Then, the uplink of a MIMO channel with multiple uer can be modeled uing (4), where H i i the E[D i ] matrix repreenting the channel repone from the cooperating node of region i to the AP, and x i repreent the E[D i ] vector of the cooperative meage ent from region i. Note that ince the node are located cloe to each other and at each intance we only conider the node which have uccefully decoded the meage, we can aume full cooperation and conider them a multiple-antenna ending the ame meage. Given the channel tate information i known at the receiver, the capacity region with multiple receive antenna can be expreed a [7] M R i (E(D i )) E H [log det(e nr + P M H i H H i )] Z i= M, M N p, i= (6) where E nr i the identity matrix and Z = [z,...,z nr ] T i the noie vector at the receiver, where we aume z i to be a Gauian RV with variance Z. Replacing M = N p reult in the expreion giving the maximum achievable um-rate. It i hown in [] that for the cae of Gauian ource with and channel matrice H i with i.i.d complex Gauian entrie with mean zero, the above um can be analytically expreed in term of Laguerre polynomial. In our ytem model, ince it i aumed that the AP i located at a height h far enough from the node, the expected power received at the AP from all ening node approximately equal P h α. Auming Rayleigh fading, the element of each matrix H i have a Gauian ditribution and are caled by the above expected power factor. Hence, for our model, H i can be written a a caled verion of a matrix H i with zero mean complex Gauian element, H i = h H i, with P normalized to. We can now α apply Theorem 2 in [] to find an analytical expreion for the network um-rate. Thi theorem tate that the capacity of a ingle-uer MIMO channel with n t tranmit and receive antenna with power contraint P total on the tranmit ide and under Rayleigh fading equal C(,n t,p total ) = log( + Ptotal n t λ) f! = (+a f)! [La f (λ)] 2 λ a f e λ dλ, where f = min(,n t ), a = max(,n t ), L a f (x) =! ex f a d x dx (e x x a f+ ) i the aociated Laguerre polynomial of order []. The author further generalize the proof and how that under the multiuer etting with M ender each having power P total, the um-rate atifie M i= R i(n t ) C(,Mn t,mp total ). In our etting, the number of tranmitter virtual antenna in each region i n t = E[D i ], the number of receiver antenna i, and the power contraint for the tranmitter within each region i P total = n t P. Thu, the achievable um-rate atifie E[Np] i= R i C(,E[N p ]E[D i ],E[N p ]E[D i ] P h ). α V. NUMERICAL ANALYSIS AND SIMULATION RESULTS In thi ection we preent numerical reult baed on the propoed analytical framework and compare them with the imulatioeult. The capacity maximization problem ha been olved numerically by changing the cooperative region radiu and finding it optimum value. We have aumed the path lo roll-off factor to be α = 4 within the planar dik a jutified in [3] for wirele network with low-lying antenna. The free pace path lo factor between the node and the AP i however, conidered to be α = 2. The two metric of interet for thi etting are the umrate and the power efficiency. To avoid the event of very cloe node, which caue the trength of the received ignal to be unlimited in our model, a minimum ditance ǫ i aumed between the node. For the unit dik with N node, Nπǫ 2 < A = i needed to guarantee that all node can be located within the dik. We aumed ǫ = 5πN. For the imulation, the capacity reult have been averaged over 25 different network topologie. In all cae, the AP i aumed to be located at height h = above the network and Z = ha been conidered. A. Effect of Cooperative Region Radiu and Number of Receive Antenna Figure 5 preent log-caled plot of the um rate for a network with N = node. The effect of different number of receive antenna baed on the capacity reult of the previou ection i alo hown. A we expect, the total number of ucceful node in decoding determine the capacity. We oberve that the curve for different number of receive antenna have the ame characteritic in term of the point where the maximum um-rate occur. The determining factor for the network um-rate i the total number of cooperative region and the number of decoding node within each region. Therefore, the optimum regioadiu i the ame for different value of. However, the increae in the number of antenna will reult in patial multiplexing which caue the capacity increae hown in the curve. Furthermore, thi figure ugget

8 Network um rate (bit/ec/hertz) = (Simulation) =5 (Simulation) = (Simulation) =2 (Simulation) = (Analyi) =5 (Analyi) = (Analyi) =2 (Analyi) Cooperative regioadiu Network um rate (bit/ec/hertz) κ= κ=2 κ= Cooparative Region Radiu Fig. 5. Network um rate for different number of antenna at the detination. =.2. that the choice of the optimal regioadiu i crucial for all value of. The maieaon for the difference between the analyi and imulation are the edge effect, the approximation ued in calculating the average um-rate, and the fact that it i not poible in general to quantify the number of active ource choen by the cheduling cheme analytically. Since node are randomly located, the actual number of ource choen by the MIS algorithm i le than the number determined by the theoretical reult. We alo conider a Rician fading model between the node and the AP. The parameter κ (o-called K-factor) i the ratio of the energy in the pecular path to the energy in the cattered path ([7], Section 2.4.2). The larger κ i the more determinitic the channel i. Figure 6 demontrate the exitence of an optimal cooperative regioadiu for different value of κ, auming equal channel power for all value of κ. By increaing the determinitic component of the channel gain, the overall um-rate decreae ince the MIMO channel no longer benefit from a rich cattering environment. The effect of determinitic part of the channel on MIMO capacity i explained in more detail in [8]. In Figure 7, the maximum network um-rate i depicted for different number of node within the network and different activity probabilitie at the node. It can be een in the figure that increaing the activity probability lead to umrate increae a expected. Alo, having more node in the network reult in more ource and, therefore, more cooperative region to be cheduled during each tranmiion. Figure 8 give more precie intuition of how the cheme work. For each activity probability, increaing the number of node reult in cooperative region with maller radii to be optimal. Thi i expected ince our cheduling algorithm only allow non overlapping cooperative region. A the number of node increae we have to pack more cooperative region and it make intuitive ene for the optimal region to be maller. Another intereting obervation i the decreae in Fig. 6. Network um rate for = 5 and different value of κ. =.2. Network um rate (bit/ec/hertz) =.2 (Simulation) =.2 (Analyi) =.5 (Simulation) =.5 (Analyi) =.2 (Simulation) p =.2 (Analyi) Number of node in the network Fig. 7. Maximum Network um rate for different number of node in the network and different activity probabilitie the optimal radiu with the increae in activity probability. For the cooperative region of node i, chooing a large value for the radiu reult in hutting down many ource that lie in the cooperative region. The capacity lo due to thi overcome the gain obtained by cooperation. Thi fact ugget that in network with high data arrival rate, chooing direct tranmiion i the optimal trategy compared to the cooperative trategy. In Figure 9 the probability of ucceful decoding at a relay veru the relay ditance from the ource i given, for three different value of r C with r opt =.5. A the figure ugget, increaing the regioadiu above the optimum value doe not further improve the ytem performance. A an example, conider a relay located at the ditance.2 from it correponding ource. For the regioadiu r opt =.5, the probability of ucceful decoding at thi relay equal.5. Thi value i almot the ame for a region with radiu r opt =. and increae to.85 for r opt =.9. However,

9 9 Optimal cooperative regioadiu p =.2 (Simulation) =.2 (Analyi) p =.5 (Simulation) =.5 (Analyi) =.2 (Simulation) =.2 (Analyi) Network um rate (bit/ec/hertz) =.2 (Simulation) =.2 (Analyi) p =.5 (Simulation) =.5 (Analyi) =.2 (Simulation) =.2 (Analyi) Number of the node within the network Decoding threhold (β) Fig. 8. Optimal cooperative regioadiu for different number of node and different activity probabilitie. Fig.. threhold. Optimal network um-rate veru different value of decoding P[SIR(r)>β] Fig r c =.5 r c =. r c = Relay ditance r from ource Probability of ucceful decoding veru relay ditance from ource. for the region with a higher radiu of.9, a we can ee from the figure, there i a very mall probability of correctly decoding for the relay located further tha opt =.5 from the ource. In thi cae, by allowing a bigger cooperation region we have allowed the amount of interference in the network to increae due to more interfering relay, while there i only little increae in the number of decoding relay which are cloe to the ource. Thi reult in the overall decreae of the um-rate compared to the cae of relaying with optimal cooperative regioadiu. B. Effect of Node Decoding Threhold The decoding threhold, β, of potential relay node can lead to noticeable change of the um-rate and energy aving within the network. In thi ection we tudy thi effect for a wide range of change in the decoding threhold. Having a mall threhold lead to the idealized cenario where the node can tolerate a high level of interference. On the other hand, a big decoding threhold will reult in the cae where cooperation cannot benefit the network by increaing it capacity or power efficiency, and direct tranmiion in both lot will be the optimal tranmiion trategy. Figure illutrate the capacity decreae a a reult of increae in the decoding threhold β. A we expect, the change in the um-rate i mall in cae where we have a higher activity probability ( =.2), which i due to the fact that the optimal olution either doe not involve cooperation or the amount of cooperation i negligible. Therefore, the change in the threhold doe not ignificantly affect the performance. However, the decreae i apparent in the low activity regime ( =.2). We have tudied the change of optimal cooperative region in Figure. For lower decoding threhold, the cooperation performance i not ignificantly affected by the amount of interference. Thi fact reult in large radiu in an optimal cooperative region. However, increaing the threhold caue the relay to be more enitive to interference. Thi lead to ineffectivene of the relay that are cloe to the boundarie of a large cooperative region, which ugget the choice of maller region a the optimal olution in cooperative communication. C. Cooperative Gain We evaluate the power efficiency by uing the notion of cooperation gain. To define thi notion more preciely, we aume that there are N active ource within the network each tranmitting with power P under the direct tranmiion etting. We call the achievable rate under thi etting R dir. We compute the overall required power under the cooperative etting to achieve the ame um-rate (i.e., we have R Tot = R dir ). Under the optimal cooperative etting, a total of N p +D node are active (where N p repreent the number of ource and D repreent the number of ucceful decoding relay), each ending with power P to achieve an overall throughput of R Tot = R dir. The cooperation gain of the network i defined

10 Optimal cooperative regioadiu =.2 (Simulation) =.2 (Analyi) =.5 (Simulation) p =.5 (Analyi) =.2 (Simulation) =.2 (Analyi) and power aving that can be obtained via cooperation. The performance gain obtained via cooperation i limited by the inherent increae in the amount of interference that the relay can caue. We tudy the optimal amount of cooperation by evaluating the trade-off between exploiting the node a relay and the increae of interference caued by aynchronou tranmiion of the relay in a dene wirele network. We introduce the notion of cooperative region, whoe radiu can be optimized to maximize the overall network um-rate. The power efficiency obtained via the choice of the optimal cooperative region i evaluated. Numerical reult baed on the propoed analyi provide deign guideline for optimal relaying in interference limited wirele network and illutrate the potential performance gain obtained by cooperation. Fig.. Cooperation gain Fig. 2. a Decoding threhold (β) Optimal cooperatioegioadiu veru the decoding threhold. =.2, N= =.2, N=25 =.5, N= =.5, N=25 =.2, N= =.2, N= Decoding threhold ( β) Cooperative gain veru the decoding threhold. Cooperation Gain = N P (N p + D)P. (7) A hown in Figure 2, the imulatioeult how ignificant power aving for low arrival network with low decoding threhold. The power aving effect decreae with the increae in the threhold or activity. In order to compare the power aving a a function of the number of node, we conider network with the ame total tranmiion power. Figure 2 how that depite increaing the number of node when uing cooperation, which reult in each node having a maller hare of power, the network cooperation gain increae. VI. CONCLUSION Thi work preent an analytical framework to tudy the effect of cooperation in large wirele network with interference mitigation. We have evaluated the potential um-rate increae APPENDIX I PROOF OF PROPOSITION II Note that D i are not independent from N p and the conditional probability reult do not apply. Wald equality [9] tate that if {D i < i < N p } are i.i.d random variable each with mean E[D i ] and N p i a topping rule for D i and D = D D Np, then E[D] = E[D i ]E[N p ]. We firt how that N p i a topping rule for D i. For any ource < i < N, if it i allowed to tranmit it mean that the cheduling algorithm permit region i to be added to the et of cooperative region. Therefore, for a new ource choen among the network node, it ha to atify thi ditance criterion. Thi criterion i only dependent on the location of other ource,...,i, which have been elected by the cheduler prior to chooing i. Therefore, N p i a topping rule, and by Wald equality E[D] = E[N p ]E[D i ]. APPENDIX II DERIVATION OF THE CDF OF SIR AT THE RELAYS The received ignal from at m ha the average power µ = P d, and therefore, the ditribution of the received power α z in a Rayleigh environment follow f Z (z) = µ e z µ. In a large network with interference, we aume that the amplitude of noie at each relay i mall relative to the interference. Therefore, it uffice to find the ditribution of SIR, Y = Z I m. Since Z and I m are poitive, the ditribution of the SIR i computed a F Y (y) = = z=xy x= z= z=xy x= z= µ e z µ f Z (z)f Im (x)dzdx x E[NI] (E[N I ] )!µ E[NI] e x µ dzdx x E[NI] = ( e yx x µ ) x= (E[N I ] )!µ E[NI] e µ dx = ( + ye[i m] ), E[NI] µ (8) where we have ued the table of integral [2] to obtain the lat equality.

11 REFERENCES [] I. E. Telatar, Capacity of multi-antenna Gauian channel, European Tranaction on Telecommunication, vol., pp , November- December 999. [2] T. M. Cover and A. A. Elgamal, Capacity theorem for the relay channel, IEEE Tranaction on Information Theory, vol. 25, pp , Sept [3] M. Gatpar and M. Vetterli, The capacity of large Gauiaelay network, IEEE Tranaction on Information Theory, vol. 5, pp , March 25. [4] J. N. Laneman and G. W. Wornell, Ditributed pace-time-coded protocol for exploiting cooperative diverity in wirele network, IEEE Tranaction Information Theory, vol. 49, pp , Oct. 23. [5] J. N. Laneman, D. N. Te, and G. W. Wornell, Cooprative diverity in wirele network: Efficient protocol and outage behaviour, IEEE Tranaction on Information Theory, vol. 5, pp , Dec. 24. [6] A. Sendonari, E. Erkip, and B. Azhang, Uer cooperation diveritypart I: Sytem decription, IEEE Tranaction on Wirele Communication, vol. 5, pp , November 23. [7] R. U. Nabar, H. Bölckei, and F. W. Kneubühler, Fading relay channel: Performance limit and pacetime ignal deign, IEEE Journal on Selected Area in Communication, vol. 22, pp. 99 9, Augut 24. [8] L. Sankaranarayanan, G. Kramer, and N. B. Mandayam, Hierarchical enor network: Capacity bound and cooperative trategie uing the multiple-acce relay channel model, in Proc. IEEE SECON, pp. 9 99, October 24. [9] B. S. Mergen, A. Scaglione, and G. Mergen, Aymptotic analyi of multi-tage cooperative broadcat in wirele network, IEEE Tranaction on Information Theory, vol. 52, pp , June 26. [] S. Vakil and B. Liang, Balancing cooperation and interference in wirele enor network, in Proc. IEEE SECON 26, vol., pp , September 26. [] T. Q. S. Quek, D. Dardari, and M. Z. Win, Energy efficiency of dene wirele enor network: To cooperate or not to cooperate, IEEE Journal on Selected Area in Communication, vol. 25, February 27. [2] D. Dardari, A. Conti, C. Buratti, and R. Verdone, Mathematical evaluation of environmental monitoring etimation error through energyefficient wirele enor network, IEEE Tranaction on Mobile Computing, pp , July 27. [3] P. Venkitaubramaniam, S. Adireddy, and L. Tong, Senor network with mobile acce: Optimal random acce and coding, IEEE Journal on Selected Area in Communication, vol. 22, pp , Augut 24. [4] P. Gupta and P. Kumar, The capacity of wirele network, IEEE Tranaction on Information Theory, vol. 46, pp , March 2. [5] M. Luby, A imple parallel algorithm for the maximal independent et problem, Journal of the ACM, vol. 5, pp , November 986. [6] R. M. de Morae, H. R. Sadjadpour, and J. Garcia-Luna-Aceve, Throughput-delay analyi of mobile ad-hoc with a multi-copy relaying trategy, in Proc. IEEE SECON 24, no. 2-29, October 24. [7] D. Te and P. Viwanath, Fundamental of Wirele Communication. Cambridge Univerity Pre, May 25. [8] S. K. Jayaweera and H. V. Poor, On the capacity of multiple-antenna ytem iician fading, IEEE Tranaction on Wirele Communication, vol. 4, May 25. [9] R. G. Gallager, Dicrete Stochatic Procee. Kluwer Academic Publiher, 996. [2] I. S. Gradhteyn and I. M. Ryzhik, Table of Integral, Serie, and Product. New York: Academic Pre, 98. Sam Vakil received B.Sc. degree in Electrical Engineering from Sharif Univerity of Technology in Tehran, Iran, in 22 and the M.Eng degree in Electrical Engineering from McGill Univerity in Montreal, Canada, in 22. He i now puruing the Ph.D degree in Electrical & Computer Engineering at the Univerity of Toronto. Hi current reearch interet include the deign and analyi of cooperative communication protocol for wirele ad-hoc and wirele network. Ben Liang received honor imultaneou B.Sc. (valedictorian) and M.Sc. degree in electrical engineering from Polytechnic Univerity in Brooklyn, New York, in 997 and the Ph.D. degree in electrical engineering with computer cience minor from Cornell Univerity in Ithaca, New York, in 2. In the 2-22 academic year, he wa a viiting lecturer and pot-doctoral reearch aociate at Cornell Univerity. He joined the Department of Electrical and Computer Engineering at the Univerity of Toronto in 22, where he i now an Aociate Profeor. Hi current reearch interet are in mobile networking and multimedia ytem. He won an Intel Foundation Graduate Fellowhip in 2 toward the completion of hi Ph.D. diertation and an Early Reearcher Award (ERA) given by the Ontario Minitry of Reearch and Innovation in 27. He wa a co-author of the Bet Paper Award at the IFIP Networking conference in 25 and the Runner-up Bet Paper Award at the International Conference on Quality of Service in Heterogeneou Wired/Wirele Network in 26. He i an editor for the IEEE Tranaction on Wirele Communication and an aociate editor for the Wiley Security and Communication Network journal. He erve on the organizational and technical committee of a number of conference each year. He i a enior member of IEEE and a member of ACM and Tau Beta Pi.